When you pull the rope on my mower it won't catch to turn the motor over?

The "recoil starter" mechanism contains a ratchet (a sort of cogged wheel and a couple of pawls to to catch the cogs). Sometimes the pawls get sticky. Remove the recoil starter (generally 4 nuts or screws) and check out the underside. You will see that if you pull slowly on the rope 2-4 pawls pop out of the ring on the underside. These pawls (which should engage the cogs on the top spindle of the motor) are what's not moving freely.

Wipe the inside of the recoil starter and the top spindle of the motor clean, and then spray/squirt a SMALL amount of oil or other lube into the slots where the pawls are. If possible, work the pawls in and out with your finger a bit to loosen them up. Pull the rope slowly again to see if the pawls are moving freely. It might take 3-4 tries of squirting lube and working the pawls.

If you can't get it working that way, it is generally possible to disassemble and repair the recoil starter, but it's a bit of a challenge. (The 5-10-foot spring will pop out as soon as you open it.) But the units are relatively cheap to replace, so I'd pick that option unless I wanted a project to keep me busy for several hours.